Portfolios: Diversification


Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University

Portfolio Returns

Consider a $100,000 portfolio with 40% invested in one asset (Asset A) and 60% in a second asset (Asset B).

  • Suppose A \(\uparrow\) 20%. $40,000 \(\rightarrow\) $48,000
  • Suppose B \(\uparrow\) 10%. $60,000 \(\rightarrow\) $66,000
  • $100,000 \(\rightarrow\) $114,000
  • So, up 14% = 0.4 \(\times\) 20% + 0.6 \(\times\) 10%

Notation

\(r_{A}\) and \(r_{B}\) denote the returns of A and B, and \(\bar{r}_{A}\) and \(\bar{r}_{B}\) denote their means (expected returns).

\(w_{A}\) and \(w_{B}\) denote portfolio weights = allocations

(like 0.4 and 0.6).

Portfolio return is \(r_{p}=w_{A}r_{A}+w_{B}r_{B}\)

In general, \(r_{p}=\sum_{i=1}^n w_ir_i\) and weights sum to 1.

The Mean Portfolio Return is . . .

\[\bar{r}_p=w_{A}\bar{r}_A+w_{B}\bar{r}_B\]

In general, \(\bar{r}_p=\sum_{i=1}^n w_i\bar{r}_i\)

Portfolio Variance

Variance means squared deviation from mean, so

\[var(r_{p})=E[(r_{p}-\bar{r}_p)^2]\]

A line of algebra shows that the deviation from the mean \(r_{p}-\bar{r}_{p}\) is a weighted average of deviations from means:

\[r_{p}-\bar{r}_{p}=w_{A}(r_{A}-\bar{r}_A)+w_{B}(r_{B}-\bar{r}_B)\]

So, the squared deviation is

\[\small w^{2}_{A}(r_{A}-\bar{r}_{A})^2+w^{2}_{B}(r_{B}-\bar{r}_{B})^2+2w_{A}w_{B}(r_{A}-\bar{r}_{A})(r_{B}-\bar{r}_{B})\]

Taking the expectation, the first two terms are variances and the third is a covariance.

Portfolio Variance with Two Assets

\[\small var(r_{p})=w^2_{A}var(r_{A})+w^2_{B}var(r_{B})+2w_{A}w_{B}cov(r_{A},r_{B})\]

  • \(var(r_{p})\) is smaller when the covariance is smaller.
  • Covariance is correlation times product of standard deviations, so \(var(r_{p})\) is smaller when the covariance is smaller.

Portfolio Variance with n Assets

\[\small var(r_{p})=\sum_{i=1}^n w_{i}^2var(r_i)+2\sum_{i=1}^n \sum_{j=i+1}^n w_{i}w_{j}cov(r_{i},r_{j})\]

There are \(\frac{n(n-1)}{2}\) covariance terms.

A Simple Case

\(n\) assets, all variances are the same = \(\sigma^2\),
all correlations are the same = \(\rho\),
all weights = \(\frac{1}{n}\).

\[\small var(r_{p})=\sum_{i=1}^n (\frac{1}{n})^2\sigma^2+2\sum_{i=1}^n \sum_{j=i+1}^n (\frac{1}{n})^2\rho\sigma^2\]

\[\small var(r_{p})=\frac{1}{n}\sigma^2+\frac{n-1}{n}\rho\sigma^2 \rightarrow \rho\sigma^2\]

3C

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2A

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Portfolio Std Dev < Weighted Average of Asset Std Devs

\[\small (w_{A}\sigma_{A}+w_{B}\sigma_{B})^2=w_{A}^2\sigma_{A}^2+w_{B}^2\sigma_{B}^2+2w_{A}w_{B}\sigma_{A}\sigma_{B}\]

same as \(var_{p}\) except no correlation in the last term. So,
\[(w_{A}\sigma_{A}+w_{B}\sigma_{B})^2>var(r_{p})\]

Some Diversification Usually Lowers Risk

Compare holding only asset B to holding some mix of A and B.

Substitute \(w_{B}=(1-w_{A})\). Variance in general is

\[\small w_{A}^2\sigma_{A}^2+(1-w_{A})^2\sigma_{B}^2+2w_{A}(1-w_{A})\rho\sigma_{A}\sigma_{B}\]

Derivative with respect to \(w_{A}\) evaluated at \(w_{A}\)=0:

\[\small 2w_{A}\sigma^2_{A}-2(1-w_{A})\sigma^2_{B}+2(1-2w_{A})\rho\sigma_{A}\sigma_{B}=-2\sigma^2_{B}+2\rho\sigma_{A}\sigma_{B}\]

Risk is lowered by \(w_{A}\) \(\uparrow\) when \(\rho\sigma_{A}<\sigma_{B}\)